Relaxation of monotone coupling conditions: Poisson approximation and beyond

Abstract

It is well-known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also give explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.MSC 2010 Primary: 62E17. Secondary: 60E15, 60F05, 62E10 IntroductionIt is well-known that in many situations exploiting negative or positive dependence structure is an effective way to establish Poisson approximation results. For example, Barbour, Holst and Janson [3] treat many applications of Poisson approximation for sums of (dependent) Bernoulli random variables X 1 , X 2 , . . . , X n which are negatively related, that is, satisfyfor all i = 1, . . . , n and increasing functions φ : {0, 1} n−1 → {0, 1}.The Poisson approximation bounds given in [3] under this negative relation assumption have the advantage of only depending on the first two moments of W . In general, such bounds require much more detailed information about the X i in order to be evaluated.

Poisson approximation with applications to stochastic geometry

“…Requiring some (stochastic) ordering assumptions between a random variable and its size-bias coupling leads to Poisson approximation results. In a similar spirit to our work, these ordering conditions were relaxed in [14]. For some recent Poisson process convergence results related to stochastic geometry we refer the reader to [28,32].…”

Poisson approximation with applications to stochastic geometry

“…Requiring some (stochastic) ordering assumptions between a random variable and its sizebias coupling leads to Poisson approximation results. In a similar spirit to our work, these ordering conditions were relaxed in [14]. For some recent Poisson process convergence results related to stochastic geometry we refer the reader to [28,32].…”